Theorem sbequ12r 2112

Description: An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)

Assertion

Ref	Expression
sbequ12r	⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))

Proof of Theorem sbequ12r

Step	Hyp	Ref	Expression
1		sbequ12 2111	. . 3 ⊢ (𝑦 = 𝑥 → (𝜑 ↔ [𝑥 / 𝑦]𝜑))
2	1	bicomd 213	. 2 ⊢ (𝑦 = 𝑥 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))
3	2	equcoms 1947	1 ⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑 ↔ 𝜑))

Syntax hints:   → wi 4   ↔ wb 196  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-sb 1881

This theorem is referenced by:  sbequ12a  2113  sbid  2114  sb5rf  2422  sb6rf  2423  2sb5rf  2451  2sb6rf  2452  opeliunxp  5170  isarep1  5977  findes  7096  axrepndlem1  9414  axrepndlem2  9415  nn0min  29567  esumcvg  30148  bj-abbi  32775  bj-sbidmOLD  32831  wl-nfs1t  33324  wl-sb6rft  33330  wl-equsb4  33338  wl-ax11-lem5  33366  sbcalf  33917  sbcexf  33918  opeliun2xp  42111